Signals are sometimes received at two or more signal receivers and are subtracted to form a single difference signal in which common errors are canceled. One problem with this approach is that the processing paths for the two or more signals may not be identical and may introduce additional, unintended differences into the signals as processed. With reference to FIG. 1, consider a signal s(t) transmitted by a single signal source 11 that is received by each of two signal antennas, 13A1 and 13A2, where the two received signals, s.sub.A1,a (t) and s.sub.A2,a (t) are pre-processed and transported by the transport paths, 15A1 (P1) and 15A2 (P2), to input terminals of respective signal receivers, 17A1 and 17A2, which are optionally parts of a single processor 19. After initial processing (amplification, smoothing, frequency downconversion, etc.), the respective signals, s.sub.A1,ro (t) and s.sub.A2,ro (t), produced at the output terminals of the two receivers, 17A1 and 17A2, are formed into a single difference signal s.sub.SD,ro (t;A1;A2)=s.sub.A1,ro (t)-s.sub.A2,ro (t), in which common errors in the two receiver output signals, s.sub.A1,ro (t) and s.sub.A2,ro (t), are canceled. These common errors may include timing errors and signal generation and transmission errors at the signal source 11, as well as other common errors. The signals s.sub.A1,ro (t) and s.sub.A2,ro (t) produced at the receiver output terminals are different from s.sub.A1,a (t) and s.sub.A2,a (t), respectively, as a result of (i) pre-processing of the received signals at the antennas, (ii) signal changes that occur during transport of the received signals, s.sub.A1,a (t) and s.sub.A2,a (t), to the respective receivers 17A1 and 17A2, (iii) signal changes that occur during processing of the received and transported signals at the respective receivers, and (iv) other distinguishing factors. Some of these distinguishing factors are approximately constant and independent of time, and other distinguishing factors will vary substantially with passage of time.
As measured at the receiver, a carrier phase observation includes additive effects of: (1) the true range RP.sub.m (t) between the location of the satellite (p) at the time the carrier phase signal was transmitted and the location of the receiver antenna (m) at the time the carrier phase signal is received; (2) an integer number of cycles NP.sub.m (t) (ignoring a fractional part f) corresponding to the distance between the satellite at transmission time and the receiver at the time t the signal is received; (3) satellite clock error .DELTA.tP(t) at the time the carrier phase signal was transmitted; (4) receiver clock error .DELTA.t.sub.m (t) at the time the carrier phase signal is received; (5) phase advance IP.sub.m (t) introduced by propagation of the signal through the ionosphere (relative to propagation of the signal through a vacuum over the same distance); (6) phase retardation or delay TP.sub.m (t) introduced by propagation of the signal through the troposphere (relative to propagation of the signal through a vacuum over the same distance); (7) time delay d.sub.m (t) introduced in transmitting and processing the carrier phase signal after this signal is received at the receiver antenna (designated as receiver line bias , due to cable lengths, signal processing delays and the like; (8) time delay dP(t) introduced in processing the carrier phase signal before this signal is transmitted by the satellite (designated as satellite line bias), due to cable lengths, signal processing delays and the like; and (9) a multipath signal contribution and random carrier phase measurement noise eP.sub.m (t) ("extraneous receiver error") at the receiver. The notation used here-is close to that adopted by A. Leick in GPS Satellite Surveying, John Wiley & Sons, Second Edition, 1995, pp. 255-264. The ionospheric time delay effect depends upon the carrier frequency f used by the satellite. The carrier phase observation .phi.P.sub.m (t) is thus expressible by the relation EQU .phi.P.sub.m (t)=RP.sub.m (t)+.lambda.NP.sub.m (t)+c.DELTA.tP(t)-c.DELTA.t.sub.m (t)-IP.sub.m (t)+TP.sub.m (t)+d.sub.m (t)+dP(t)+eP.sub.m (t). (1)
where c is the velocity of light in a vacuum and .lambda. is the carrier wavelength. The phase difference .phi.P.sub.m (t)=.phi..sub.m (t)-.phi.P(t) is usually developed for propagation in a vacuum, and terms, such as IP.sub.m (t) and TP.sub.m (t), are added to account for propagation in a medium other than a vacuum.
Where vehicle attitude or angular orientation parameters, such as yaw, pitch and/or roll of a land, waterborne, airborne or spaceborne vehicle, are to be determined, two or more spaced apart antennas are often located on the vehicle, and GPS signals are received and processed from the same group of satellites at each of these antennas. The baseline length between any two of these antennas is necessarily relatively small, usually no more than 100 meters and often as small as 1-2 meters.
In single difference carrier phase observations, carrier phase signals received at each of two spaced apart antennas are subtracted from each other to reduce or eliminate errors and other effects that are common to the phase observations measured at both receivers; these common errors include the satellite clock errors and satellite processing delays. A single difference phase observation is defined by the relations EQU .phi.P.sub.m,n (t)=.phi.P.sub.m (t)-.phi.P.sub.n (t)=RP.sub.m,n (t)+.lambda.NP.sub.m,n (t)-c.DELTA.t.sub.m,n (t)+d.sub.m,n(t)+eP.sub.m,n (t), (2) EQU RP.sub.m,n (t)=RP.sub.m (t)-RP.sub.n (t), (3) EQU NP.sub.m,n (t)=NP.sub.m (t)-NP.sub.n (t), (4) EQU .DELTA.t.sub.m,n (t)=.DELTA.t.sub.m (t)-.DELTA.t.sub.n (t), (5) EQU d.sub.m,n (t)=d.sub.m (t)-d.sub.n (t), (6) EQU eP.sub.m,n (t)=eP.sub.m (t)-eP.sub.n (t). (7)
The nearly common error due to ionospheric phase advance and the nearly common error due to tropospheric phase delay are dropped in Eq. (2), based on two arguments: (1) these phase advance or delay errors are nearly the same for two adjacent antennas, and thus they cancel when subtracted from each other; and (2) the ionospheric phase advance and tropospheric phase delay can be separately modeled and estimated for each receiver and the common satellite and thus can be separately removed from Eq. (2).
Conventional carrier phase analysis based on the single difference carrier phase observation in Eq. (1) requires determination or cancellation of the respective clock errors .DELTA.t.sub.m (t) and .DELTA.t.sub.n (t), which can be performed using clock measurements with an atomic clock, such as a Cs or Rb clock, or use of the same clock for both receivers. Determination of the receiver line biases d.sub.m (t) and d.sub.n (t) is not so straightforward, because each of the variables depends upon independent variables such as receiver cable lengths, method(s) of processing the signals received, receiver age, receiver component(s) bias and drift, local temperature and other imponderables.
"Line bias" in a two-antenna system is the difference of signal changes, due to different transport paths and different processing devices and/or methods, from the antennas to a point at which a difference of the two received signals is formed, called a single difference signal. A single difference observation SD(t;j;A1;A2) from a signal source j includes single difference observations at the two antennas, plus contributions from line bias, receiver noise and multipath signals. One problem here is to separate line bias from the remainder of a single difference observation.
Line biases can be canceled by using a second signal source and forming double differences for the two signal sources and the two receivers. However, the double difference approach sacrifices one measurement and increases the statistical error level (theoretically by a multiplicative factor of 1.414, assuming white noise is present) arising from multipath signals and receiver noise.
What is needed is a method and system for compensating for line biases without using a double difference method. The approach should be capable of quantitatively estimating variation of line bias around a well-defined initial line bias in approximately real time and should allow for changes in line bias variation with the passage of time.